Remark  Lagrangians of the form \(\hat{L}=\frac{g({x}_t,{x}_t)}{\alpha ({x}_t)}\)  for a Riemannian or pseudo-Riemannian metric \(g\)  and for a
1-form \(\alpha \)  are called Kropina (pseudo-Finsler) metrics. Kropina metrics were also considered in the framework of mathematical relatively,
see e.g. E. Caponio et al [1]}.
Kropina metrics are not defined on the vectors \({x}_t\)  lying in the kernel of \(\alpha \) . Note that in our case the form \(\alpha \)  is closed
so its extremals define a projective structure by [2]}.

